方兆本 随机过程 答案.pdf

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1、n:^!G90

2、-W9'>1XVjfie11.R{X(t),t∈T}
Y%3;F1}.,(6pFC{*CE[X(s)]ÆE[X(s)X(s+t)]}As.op}.F*,{X(t),t∈T}pC{*CZ>!A,G:(1)µX(s)=E[X(s)]s∈T'&;(2)RX(t,s)=Cov[X(t),X(s)]0Æt−sx.;DCov[X(t),X(s)]=E[X(t)X(s)]−E[X(t)]E[X(s)],8,>!(1)(2)ÆZ>!H:(3)µX(s)=E

3、[X(s)]s∈T'&;(4)E[X(t)X(s)]0Æt−sx.#1(d{X(t),t∈T}pC{*CE[X(s)]ÆE[X(s)X(s+t)]}As.2.U1,···,Un
(0,1)57bFnnG1.V0t,Pn1X(t)=nI(t,Uk),0≤t≤1,#U1,···,UnF.wb&.}1}.k=1{X(t),0≤t≤1}F7-Bf_%&.[:*,{X(t),0≤t≤1}F7-&
Xn1µX(t)=E[X(t)]

4、=E[I(t,Uk)]nk=1=E[I(t,U1)],0≤t≤1,(1)Bf_%&
RX(t,s)=Cov[X(t),X(s)]]1XnXn=Cov[I(t,Uk),I(s,Ul)]n2k=1l=1Xn1=Cov[I(t,Uk),I(s,Uk)]n2k=11=Cov[I(t,U1),I(s,U1)],0≤t≤1,(2)nXE[I(t,U1)]=P(U1≤t)=t,0≤t≤1,(3)2n:^!G90

5、-W9'>E[I(t,U1)I(s,U1)]=P(U1≤t,U1≤s)=P(U1≤min{t,s})=min{t

6、,s},0≤t,s≤1,Cov[I(t,U1),I(s,U1)]=E[I(t,U1)I(s,U1)]−E[I(t,U1)]E[I(s,U1)]=min{t,s}−ts,0≤t,s≤1,(4)"(3)(4)(1)(2)5EµX(t)=t,0≤t≤1,1RX(t,s)=[min{t,s}−ts],0≤t,s≤1.n3.2RZ1,Z2
GAbF'81,7-
0,_%
σ,λ
&.X(t)=Z1cos(λt)+Z2sin(λt).}{X(t),t∈(−∞,+∞)}F7-&Bf_%&.6pFZ?

7、[:<*,{X(t),t∈(−∞,+∞)}F7-&
µX(t)=E(Z1)cos(λt)+E(Z2)sin(λt)=0,t∈(−∞,+∞),Bf_%&
RX(t,s)=Cov[Z1cos(λt)+Z2sin(λt),Z1cos(λs)+Z2sin(λs)]=Cov(Z1,Z1)cos(λt)cos(λs)+Cov(Z1,Z2)cos(λt)sin(λs)+Cov(Z2,Z1)sin(λt)cos(λs)+Cov(Z2,Z2)sin(λt)sin(λs)22=σcos(λt)cos(λs)+σsin(λt)sin(

8、λs)2=σcos[λ(t−s)],t,s∈(−∞,+∞).w{X(t),t∈(−∞,+∞)}pF.4.Poisson}.{X(t),t≥0}[F(i)X(0)=0;(ii)Vt>s,X(t)−X(s)f:7-
λ(t−s)FPoissonb;(iii)}.GF.}r7-&Bf_%&.6pFZ?[{X(t),t≥0}F7-&
µX(t)=E[X(t)]=E[X(t)−X(0)]=λt,t>0,{X(t),t≥0}F_%&
Var[X(t)]=Var[X(t)−X(0)]=λt,t>0,Vs>t>

9、0,E[X(t)X(s)]=E{[X(t)−X(0)][(X(s)−X(t))+(X(t)−X(0))]}n:^!G90

10、-W9'>32=E{[X(t)−X(0)]}+E{[X(t)−X(0)][X(s)−X(t)]}2=Var[X(t)−X(0)]+{E[X(t)−X(0)]}+E[X(t)−X(0)]E[X(s)−X(t)]2=λt+(λt)+λt·λ(s−t)=λt(λs+1),8,{X(t),t≥0}FBf_%&
RX(t,s)=E[X(t)X(s)]−E[X(t)]E[X(s)]=λt,s≥t>0

11、,B`x&
RX(t,s)rX(t,s)=[RX(t,t)RX(s,s)]1/2rt=,s≥t>0.s5.{X(t),t≥0}
*:5FPoisson}..Y(t)=X(t+1)−X(t),}}.{Y(t),t≥0}F7-&Bf_%&,u0rpm.[{Y(t),t≥0}F7-&
µY(t)=E[X(t+1)]−E[X(t)]=µX(